Contents

# Contents

## Idea

For $X$ a topological space and $n \in \mathbb{N}$ a natural number, the space of finite subsets of cardinality $\leq n$ in $X$ is the suitably topologized set of finite subsets $S \subset X$ of cardinality $\left\vert S\right\vert \leq n$, often denoted $\exp^n X$ or similar (e.g. Félix-Tanré 10).

## Properties

### Relation to unordered configuration space of points

The topological subspace of finite subsets of cardinality exactly equal to $n$ is the unordered configuration space of points in $X$

(1)$Conf_n(X) \hookrightarrow \exp^n(X)$
###### Proposition

Let $X$ be an non-empty regular topological space and $n \geq 2 \in \mathbb{N}$.

Then the injection (1)

(2)$Conf_n(X) \hookrightarrow \exp^n(X)/\exp^{n-1}(X)$

of the unordered configuration space of n points of $X$ into the quotient space of the space of finite subsets of cardinality $\leq n$ by its subspace of subsets of cardinality $\leq n-1$ is an open subspace-inclusion.

Moreover, if $X$ is compact, then so is $\exp^n(X)/\exp^{n-1}(X)$ and the inclusion (2) exhibits the one-point compactification $\big( Conf_n(X) \big)^{+}$ of the configuration space:

$\big( Conf_n(X) \big)^{+} \;\simeq\; \exp^n(X)/\exp^{n-1}(X) \,.$